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Beliebt Rechnen >

integral of (x^2+5x)e^{x/3}

Lösung

\int (x^{2} + 5 x) e^{\frac{x}{3}} d x

Lösung

3 e^{\frac{x}{3}} x^{2} - 3 e^{\frac{x}{3}} x + 9 e^{\frac{x}{3}} + C

Schritte zur Lösung

\int (x^{2} + 5 x) e^{\frac{x}{3}} d x

Multipliziere aus

(x^{2} + 5 x) e^{\frac{x}{3}} : e^{\frac{x}{3}} x^{2} + 5 e^{\frac{x}{3}} x

= \int e^{\frac{x}{3}} x^{2} + 5 e^{\frac{x}{3}} x d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int e^{\frac{x}{3}} x^{2} d x + \int 5 e^{\frac{x}{3}} x d x

\int e^{\frac{x}{3}} x^{2} d x = 3 e^{\frac{x}{3}} x^{2} - 6 (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}})

\int 5 e^{\frac{x}{3}} x d x = 5 (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}})

= 3 e^{\frac{x}{3}} x^{2} - 6 (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}}) + 5 (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}})

Vereinfache

= 3 e^{\frac{x}{3}} x^{2} - 3 e^{\frac{x}{3}} x + 9 e^{\frac{x}{3}}

Füge eine Konstante zur Lösung hinzu

= 3 e^{\frac{x}{3}} x^{2} - 3 e^{\frac{x}{3}} x + 9 e^{\frac{x}{3}} + C

Graph

Beliebte Beispiele

integral of (5^{x+1}-2^{x+1})/(10^x)

\int \frac{5 ^{x + 1} - 2 ^{x + 1}}{1 0 ^{x}} d x

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integral of-6x^5e^{x^6}

\int - 6 x^{5} e^{x^{6}} d x

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\int \frac{1}{1 - 2 x + 3} d x